174 research outputs found
Mirror Prox Algorithm for Multi-Term Composite Minimization and Semi-Separable Problems
In the paper, we develop a composite version of Mirror Prox algorithm for
solving convex-concave saddle point problems and monotone variational
inequalities of special structure, allowing to cover saddle point/variational
analogies of what is usually called "composite minimization" (minimizing a sum
of an easy-to-handle nonsmooth and a general-type smooth convex functions "as
if" there were no nonsmooth component at all). We demonstrate that the
composite Mirror Prox inherits the favourable (and unimprovable already in the
large-scale bilinear saddle point case) efficiency estimate of
its prototype. We demonstrate that the proposed approach can be naturally
applied to Lasso-type problems with several penalizing terms (e.g. acting
together and nuclear norm regularization) and to problems of the
structure considered in the alternating directions methods, implying in both
cases methods with the complexity bounds
Semi-smooth Newton methods for mixed FEM discretizations of higher-order for frictional, elasto-plastic two-body contact problems
International audienceIn this article a semi-smooth Newton method for frictional two-body contact problems and a solution algorithm for the resulting sequence of linear systems are presented. It is based on a mixed variational formulation of the problem and a discretization by finite elements of higher-order. General friction laws depending on the normal stresses and elasto-plastic material behavior with linear isotropic hardening are considered. Numerical results show the efficiency of the presented algorithm
Gap functions for quasi-equilibria
An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimate of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm
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